\[Y = 2 + 3X + \epsilon, \epsilon \sim N(0,3)\]
\[Y = 2 + 3X + 2X^2 + \epsilon\]
\[\epsilon_i = \sigma^2\]
\[Y = 2 + 3X + \epsilon, \epsilon \sim N(0,1)\]
\[Y = 2 + 3X + \epsilon, \epsilon \sim N(0,\frac{X}{2})\]
| Transformation | Power | \(f(X)\) |
|---|---|---|
| Cube | 3 | \(X^3\) |
| Square | 2 | \(X^2\) |
| Identity | 1 | \(X\) |
| Square root | \(\frac{1}{2}\) | \(\sqrt{X}\) |
| Cube root | \(\frac{1}{3}\) | \(\sqrt[3]{X}\) |
| Log | 0 (sort of) | \(\ln(X)\) |
One-sided transformation of \(Y\)
\[\ln(Y_i) = \beta_0 + \beta_{1}X_i + \epsilon_i\]
\[E(Y) = e^{\beta_0 + \beta_{1}X_i}\]
\[\frac{\vartheta E(Y)}{\vartheta X} = e^{\beta_1}\]
One-sided transformation of \(X\)
\[Y_i = \beta_0 + \beta_{1} \ln(X_i) + \epsilon_i\]
\[\ln(Y_i) = \beta_0 + \beta_{1} \ln(X_i) + \dots + \epsilon_i\]
Elasticity
\[\text{Elasticity}_{YX} = \frac{\% \Delta Y}{\% \Delta X}\]
A double multiplicative relationship
\[y_i = \beta_0 + \beta_{1}x_{i} + \epsilon_{i}\]
\[y_i = \beta_0 + \beta_{1}x_{i} + \beta_{2}x_i^2 + \beta_{3}x_i^3 + \dots + \beta_{d}x_i^d + \epsilon_i\]
\[\text{Biden}_i = \beta_0 + \beta_1 \text{Age} + \beta_2 \text{Age}^2 + \beta_3 \text{Age}^3 + \beta_4 \text{Age}^4\]
| (Intercept) | I(age^1) | I(age^2) | I(age^3) | I(age^4) | |
|---|---|---|---|---|---|
| (Intercept) | 620.00316 | -56.31558 | 1.76432 | -0.02291 | 0.00011 |
| I(age^1) | -56.31558 | 5.20765 | -0.16556 | 0.00218 | -0.00001 |
| I(age^2) | 1.76432 | -0.16556 | 0.00533 | -0.00007 | 0.00000 |
| I(age^3) | -0.02291 | 0.00218 | -0.00007 | 0.00000 | 0.00000 |
| I(age^4) | 0.00011 | -0.00001 | 0.00000 | 0.00000 | 0.00000 |
\[\hat{f}(x_0) = \hat{\beta}_0 + \hat{\beta}_1 x_{0} + \hat{\beta}_2 x_{0}^2 + \hat{\beta}_3 x_{0}^3 + \hat{\beta}_4 x_{0}^4\]
\[\text{Var}(\hat{f}(x_o))\]
\[\Pr(\text{Voter turnout} = \text{Yes} | \text{mhealth}) = \frac{\exp[\beta_0 + \beta_1 \text{mhealth} + \beta_2 \text{mhealth}^2 + \beta_3 \text{mhealth}^3 + \beta_4 \text{mhealth}^4]}{1 + \exp[\beta_0 + \beta_1 \text{mhealth} + \beta_2 \text{mhealth}^2 + \beta_3 \text{mhealth}^3 + \beta_4 \text{mhealth}^4]}\]
Binning
\[y_i = \beta_0 + \beta_1 C_1 (x_i) + \beta_2 C_2 (x_i) + \dots + \beta_K C_K (x_i) + \epsilon_i\]
\[y_i = \beta_0 + \beta_1 x_i + \beta_2 x_i^2 + \beta_3 x_i^3 + \epsilon_i\]
Piecewise cubic polynomial with 0 knots
\[y_i = \beta_0 + \beta_1 x_i + \beta_2 x_i^2 + \beta_3 x_i^3 + \epsilon_i\]
Piecewise cubic polynomial with 1 knot
\[y_i = \begin{cases} \beta_{01} + \beta_{11}x_i^2 + \beta_{21}x_i^2 + \beta_{31}x_i^3 + \epsilon_i & \text{if } x_i < c \\ \beta_{02} + \beta_{12}x_i^2 + \beta_{22}x_i^2 + \beta_{32}x_i^3 + \epsilon_i & \text{if } x_i \geq c \end{cases}\]
\[y_i = \beta_0 + \beta_{1} X_{i1} + \beta_{2} X_{i2} + \dots + \beta_{p} X_{ip} + \epsilon_i\]
\[y_i = \beta_0 + \sum_{j = 1}^p f_j(x_{ij}) + \epsilon_i\]
\[y_i = \beta_0 + f_1(x_{i1}) + \beta_{2} f_2(x_{i2}) + \dots + f_p(x_{ip}) + \epsilon_i\]
\[\text{Biden} = \beta_0 + f_1(\text{Age}) + f_2(\text{Education}) + f_3(\text{Gender}) + \epsilon\]